This thesis analyzes two regression models in which their respective least squares estimators have nonstandard asymptotics. It is divided in an introduction and two parts. The introduction motivates the study of nonstandard problems and presents an outline of the contents of the remaining chapters. In part I, the least squares estimator of a multivariate convex regression function is studied in great detail. The main contribution here is a proof of the consistency of the aforementioned estimator in a completely nonparametric setting. Model misspecification, local rates of convergence and multidimensional regression models mixing convexity and componentwise monotonicity constraints will also be considered. Part II deals with change-point regression models and the issues that might arise when applying the bootstrap to these problems. The classical bootstrap is shown to be inconsistent on a simple change-point regression model, and an alternative (smoothed) bootstrap procedure is proposed and proved to be consistent. The superiority of the alternative method is also illustrated through a simulation study. In addition, a version of the continuous mapping theorem specially suited for change-point estimators is proved and used to derive the results concerning the bootstrap.